MIND EXPERIMENT: Remove Constant \(c\)

Purpose

This study is a deliberate removal of one assumption from the investigation of electromagnetic propagation: the EM signal has a constant propagation speed.

The purpose is not to establish that the speed of electromagnetic propagation changes. The purpose is to remove constancy as a premise and examine what the observed behavior of electromagnetic signals actually requires.

The question is therefore deliberately open:

Does the EM signal itself have a constant propagation speed, or does its propagation change with the conditions of the fields it traverses?

The Assumption

The symbol \(c\) is commonly used to represent the propagation speed of electromagnetic radiation in vacuum.

In this mind experiment, the statement "the EM signal has a constant propagation speed" is temporarily removed from the premises.

This does not make \(c\) variable by declaration. It simply prevents the investigation from using constancy of \(c\) as an assumption before examining the physical conditions under which propagation occurs.

Begin With the Signal

Consider an electromagnetic signal emitted by a source and received by a detector.

The detector records measurable properties of the arriving signal: frequency, phase, amplitude, polarization, direction, and arrival time.

The first question is therefore not:

At what constant speed did the signal travel?

Instead:

What propagation behavior is actually indicated by the measurements?

The distinction is small in wording but large in consequence. The first question assumes the property being investigated. The second leaves it open.

Propagation Through a Physical Environment

An electromagnetic signal does not propagate through an abstract equation. It propagates through whatever physical electromagnetic conditions exist between source and detector.

If those conditions are spatially or temporally different, the possibility must be considered that the propagation characteristics of the signal may also be different.

A useful working representation is therefore

\[ c=c(\mathbf{x},t) \]

This expression is not a conclusion. It is a placeholder for the question being investigated: is propagation speed actually independent of position and physical conditions?

The Transmission-Line Question

Engineering provides a useful way to examine the question without first deciding the cosmological answer.

In a distributed transmission system, propagation depends upon the electrical properties of the system.

\[ v=\frac{1}{\sqrt{LC}} \]

The characteristic impedance is correspondingly

\[ Z_0=\sqrt{\frac{L}{C}}. \]

These equations do not demonstrate that free-space propagation behaves as a transmission line. They establish an engineering question: when propagation occurs through a physical system whose electromagnetic properties vary, what happens to the propagation itself?

If an analogous description applies to the electromagnetic substrate, then local propagation could be represented by local electromagnetic parameters:

\[ c(\mathbf{x})= \frac{1}{\sqrt{\mu(\mathbf{x})\varepsilon(\mathbf{x})}}. \]

Again, this is treated here as a working relationship to investigate, not as an established description of physical space.

Remove the Constant

Once constant \(c\) is removed as a premise, a number of questions become available that would otherwise be suppressed by the initial assumption.

Does propagation speed depend upon the electromagnetic conditions encountered along the path?
Can a spatial gradient in propagation conditions produce a measurable change in signal velocity?
Can propagation conditions alter phase without destroying coherence?
Can propagation conditions alter frequency, wavelength, or both?
Can a change in propagation velocity produce an apparent timing difference without requiring time itself to change?
Can different propagation environments produce different measured values of \(c\)?

The Measurement Problem

A measurement of the speed of a signal necessarily involves more than the signal itself.

At minimum, there is a source, a propagation path, a detector, and a reference for distance and time.

\[ \text{Source} \rightarrow \text{Propagation} \rightarrow \text{Detector} \]

The measured value is therefore a property of an experimental arrangement. The question is whether the resulting value represents a universal property of electromagnetic propagation or a local property of the conditions under which the measurement was made.

This distinction becomes especially important when measurements are compared across substantially different physical environments.

Propagation Velocity and Time

Removing constant \(c\) also separates two quantities that are often treated as though they were interchangeable: propagation rate and time rate.

If a signal travels a distance \(d\) with local propagation velocity \(c(\mathbf{x})\), its propagation time may be represented as

\[ t_{\mathrm{prop}} = \int_{\mathrm{path}} \frac{dl}{c(\mathbf{x})}. \]

Under a constant-speed assumption this reduces to the familiar relationship

\[ t_{\mathrm{prop}}=\frac{d}{c}. \]

The mind experiment asks whether the second expression is a fundamental physical requirement or a special case of the first.

What Happens to Frequency?

If propagation conditions can change, then frequency becomes another quantity that must be examined rather than automatically attributed to source motion.

The familiar wave relationship is

\[ v=f\lambda. \]

If \(v\) is allowed to vary, then a change in propagation conditions can potentially alter the relationship between frequency and wavelength.

This creates a direct connection to the separate investigation of cosmological redshift.

The observation remains:

\[ f_{\mathrm{observed}} \neq f_{\mathrm{reference}}. \]

The cause remains an open question.

The Reactance Question

The removal of constant \(c\) also makes the electromagnetic properties of the propagation environment relevant to the investigation.

In an engineering system, reactance represents energy storage and therefore participates directly in the phase behavior of an alternating electromagnetic system.

This raises a question rather than providing an answer:

Could changes in the reactive condition of the propagation environment produce measurable changes in the behavior of an electromagnetic signal?

If so, phase, impedance, coupling, propagation velocity, reflection, refraction, and frequency response may all become relevant measurements of the propagation environment.

Relation to Gravity

A variable propagation velocity also creates a possible connection to the separate investigation of gravity.

If a local propagation field can be represented by \(c(\mathbf{x})\), then a spatial gradient can be examined:

\[ \nabla c(\mathbf{x}). \]

The important point for this mind experiment is not the particular gravitational relationship that might eventually be assigned to this gradient. It is that the possibility of a propagation gradient can be investigated without first assuming that \(c\) is globally constant.

Gravity, redshift, timing, and propagation can therefore be treated as potentially related observations while keeping their mechanisms separate until evidence connects them.

What the Mind Experiment Does Not Claim

This study does not claim that the speed of light has been shown to vary.

It does not establish that \(c(\mathbf{x},t)\) is the correct physical description of electromagnetic propagation.

It does not establish a physical electromagnetic substrate.

It does not establish that reactance causes redshift.

It does not establish that a propagation gradient causes gravity.

Its purpose is narrower: remove the assumption of constant propagation speed and determine what the observations require without that assumption.

Working Questions

What experimental evidence establishes the constancy of electromagnetic propagation speed under different physical conditions?
Are measurements of \(c\) local measurements, or can they establish a property independent of the measurement environment?
Can propagation through a structured electromagnetic environment produce measurable changes in phase or arrival time?
Can propagation conditions alter the observed frequency of a coherent signal?
Can an independent measurement distinguish a change in propagation velocity from a change in the physical process used to measure time?
If propagation velocity varies, what measurable electromagnetic property determines that variation?
Can such a property be measured independently of the effects it is proposed to explain?

Current Status

Study / Mind Experiment.

Constant propagation speed is removed here as an investigative premise. No alternative propagation law is asserted.

The working notation \(c(\mathbf{x},t)\) is used only to keep the possibility open while the physical evidence is examined.

Conclusion

The purpose of this mind experiment is not to replace one assumption with its opposite.

It is to remove the assumption that the EM signal must have a constant propagation speed and then examine the signal, the propagation path, and the measurement independently.

If constant \(c\) is required by the observations, the assumption survives the experiment.

If it is not required, then propagation must be investigated as a physical process whose behavior may depend upon the conditions through which the signal travels.

The objective is to determine whether the EM signal itself is a constant speed or changes with the conditions of the fields it traverses.